Question

one side of a ravine is 51 ft long, and the other is 43 ft long. an 84 ft zipline runs from the top of one side of the ravine to the other. what is the angle of the ravine?
ravine’s angle = ?°
round your answer to the nearest hundredth.

Explanation:

Step1: Identify the Law to Use

We have a triangle with sides \( a = 51 \), \( b = 43 \), and \( c = 84 \). To find the angle between sides \( a \) and \( b \) (the angle of the ravine), we use the Law of Cosines: \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), where \( C \) is the angle we want to find.

Step2: Rearrange the Formula for \( \cos(C) \)

Rearranging the Law of Cosines to solve for \( \cos(C) \):
\( \cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab} \)

Step3: Substitute the Values

Substitute \( a = 51 \), \( b = 43 \), and \( c = 84 \):
\( \cos(C)=\frac{51^{2}+43^{2}-84^{2}}{2\times51\times43} \)
Calculate the numerator: \( 51^{2}=2601 \), \( 43^{2}=1849 \), \( 84^{2}=7056 \)
Numerator: \( 2601 + 1849 - 7056 = 4450 - 7056 = -2606 \)
Denominator: \( 2\times51\times43 = 4386 \)
So, \( \cos(C)=\frac{-2606}{4386}\approx - 0.59416 \)

Step4: Find the Angle

Take the arccosine of \( -0.59416 \):
\( C=\arccos(-0.59416)\approx 126.42^{\circ} \) (using a calculator, ensuring it is in degree mode)

Answer:

\( 126.42 \)